On modifying constructed normal numbers

A NNALES DE LA FACULTÉ DES SCIENCES DE T ^OULOUSE

B ^ODO V ^OLKMANN

On modifying constructed normal numbers

Annales de la faculté des sciences de Toulouse 5

^e

série, tome 1, n

^o

3 (1979), p. 269-285

<http://www.numdam.org/item?id=AFST_1979_5_1_3_269_0>

© Université Paul Sabatier, 1979, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

ON MODIFYING CONSTRUCTED NORMAL NUMBERS

Bodo Volkmann (1)

Annales Faculté des Sciences Toulouse

Vol I,1979,

P. 269 à 285

(1 )

Mathematisches Institut

A,

Universität

Pfaffenwaldving 57,

^{D 7000}

Stuttgart

^{80 -}

Allemagne

^Fédérale.

Resume : Soient

p~

^... des entiers

positifs,

^{ecrits en base}

g > 2,

tels que le nombre reel ^{a =}

p2

^...

soit normal. On démontre _que

c~* = p * ... ,

^où

p~ = ^p~

⁺

^{j. ,}

^E

^{IN*, log} j~

⁼

^o(log p.),

^{est aussi}

^normal,

théorème

qui

^{résout un}

problème pose

par T.

SALAT.

Parmi d’autres

résultats,

^{on donne une}

generalisation

^aux

mesures arbitraires de distribution

g-adique, x,

^telles que x

( ~ 1 ~ )

^{= 0.}

Summary :

^If

p~

^{... are natural}

^numbers,

^{written to base}

^{g > 2,}

such that the real number

Q

g

g .

1

1

2

^{... IS normal, It IS proved t at Q}

g

g .

1 2

^Wit

i

⁼

i i

i ^’

i

i ^{’ og}

i

i ^{= 0 og}

pi),

^{IS a so}

normal. This theorem solves a

problem proposed by

^T.

SALAT. Among

^other

results,

^a

generalisation

^to

arbitrary g-adic

distribution measures x with x

( ~ 1 ~ )

^{= 0 is}

^given.

1. PROBLEM.

Let

p2,

^{... be an unbounded sequence} of natural numbers which

diverges

^to

infinity

^{and let}

B1 ~ B~,

^{... be the}

corresponding

^{blocks of}

digits

^{relative to a fixed}

(integral)

^base

g > 2 ; furthermore,

^we suppo-

se that the real number

(1 ) )

be normal. Mr T.

SALAT

has raised the

question (2)

whether the number

a*,

obtained in the same way but with each

pi being replaced by pi = pi

⁺

1,

is also normal.

(1 ) )

^The

pre-index

g denotes a

g-adic expansion.

(2)

Communicated to the author

by

Mr F. SCHWEIGER.

This

question

appears ^to be well motivated inasmuch as the answer is affirmative in the «classical»

cases of normal numbers constructed in the form

(1) :

a)

^For

Champernowne’s

^normal

number ^{(cf. [1] ] ),}

defined with

g =10

^and

pi

⁼

^i,

the modified num-

*

ber is a = 10

a -1, normality being

^obvious.

b)

For the normal numbers constructed

by

the method of DAVENPORT and

ERDOS _{[3] ,}

^where

p. = p(i)

^{for any} non-constant

polynomial

p which maps

IN*

into

(1 ),

^{the modified}

^{number a*}

^{satisfies the}

same

assumptions

^{and is} therefore normal.

c) Similary, normality

^of

Q *

follows

immediately

^{if a} is constructed

by

the method of COPELAND and

ERDOS [2]

which allows

(p;)

^{to be any} sequence with ^more than

n 1- E

elements not

exceeding

^{n for} every

E > 0 and all n >

no(E ^),

^{inasmuch as}

^{again a*}

satisfies the same condition.

It is the main _purpose of the

present

paper ^to show that the ^{answer to the} stated

question, in aslight- ly

^more

general form,

^is

always

affirmative

(Theorem 1).

^More

generally,

^{it will be shown} for which distribution

measures x,

assuming

^{that a have the}

digit

distribution x, the same is

always

^{true for}

a* (Theorem 4).

^Further-

more, ^we

give

^some classes of

counter-examples

which show that in Theorem 1

normality

^{cannot be}

replaced by simple normality.

^In

particular,

Theorem 3 describes all

possible pairs

^of

digit frequency

^{vectors which the}

pair

of

numbers a,

^a may possess.

The work contained in this paper ^was done

during

the author’s stay ^at Bordeaux in the

spring

^of

1978,

and he wishes to express his thanks ^to the U.E.R. de

Mathématiques

^et

d’l nformatique

for all the

hospi- tality

^and

cooperation

^received.

2. NOTATION.

2.1. - With _every

integer

^{m E fN we associate the block B =}

l3(m)

^{of its}

g-adic digits,

^{written in the cus-}

tomary ^{order. We} consider the set

Big

^{of all}

g-adic blocks, including

^the empty

block,

^{to be denoted}

by F.,

^{as a}

semi-group

^with respect ^to

juxtaposition,

^thus

regarding j3

^{as a}

mapping

^{from IN into}

Bg.

Let Bog

be the subset of obtained

by omitting F03C6

and all blocks of

lengths >

^{1 which}

begin

with the

digit

^{0. If}

~ g

^is

replaced by ,~g,

^the

^{mapping @}

^becomes

^bijective.

2.2. - For _any block F ^we denote the

length by

^{II F}

II , defining

^II

F~

^{II = 0.}

2.3. - Blocks of ^the form F F ^... F

(n times)

^{shall be denoted}

by F~n).

^{If F is the}

single digit g-l,

^we

shall also write

(g-1 )~n)

⁼

(1 )

^{We denote the set of natural number}

^by (N*, letting (N* U ~ 0 ~ _

2.4. - If

F ~ F~

^{we let}

^p(F)

^{and a}

^(F), respectively,

^{be the}

largest integers ~

0 such that F can be written in the form F ^{= =}

F"(g-1 )(o ) (see 2.3) ^F,

^with

^F’,

^F"

g (where ^evidently

^{either F’ = F or}

F" = F).

2.5. - For _any block F ~

F03C6,

^{and any} real number

x E (0,1] let AF(x,n)

denote the num-

ber of

copies

^{of F}

occurring

within the first ⁿ

digits

^{of the}

(non-terminating) ^g-adic expansion

^{of x, with}

AF _~ ^(x,n)

^{= n.}

^Similary,

^{if let} denote the number of

copies

^{of F which are embedded in B.}

^Thus, normality

^{of a} may be

expressed by

^{means of the}

equations :

The number ^a is called k-normal if

(2)

is satisfied for all F with 11 F

II = k,

^and

1-normality

^is

usually

^called

simple normality.

2.6. -

Any

block F E

~g

determines the «basic interval» of numbers x

E ( 0,1 ] whose ^g-adic

^expan-

sion

begins

with F. For the sake

of simplicity

^we shall denote this interval

by F also, letting F~=(0,1 ].

2.7. - For _any set L c

IN

let

L(n)

denote the number of elements not

exceeding

ⁿ

(n = 1,2,...).

3. MAIN RESULT.

We can now state our main result :

THEOREM 1. Let

pi, Bi ⁽ⁱ

⁼

^1,2,...)

^{and a} be defined as in

(1),

^let

Jl,l2,"’,~i

^E

^IN,

be numbers such that

log j.

⁼

o(log p.), p. = _{P ;}

⁼

^Bi (i

⁼

1,2,...).

^Then

normality

^{of a}

implies normality

of the number

a)

We define the numbers M

B;

^{!! =}

q;,

^{!! !! =}

q~ ^s~

⁼

^q~

^{+ ... +}

^q;, s~=

^{+ ... +}

^z~

^{= !!}

j~(j;)

^!!

and the sets S ⁼

~ ,s~,... ~ ^S* = ~ s~ ,s~... ~,

^thus

^having

Consequently,

^since

p. i -~

^{we obtain}

s. i+1 - s. i =

^{1 7 ~ and}

s* i+1 - s/-

^which

^implies

^that

^{(1 )}

~)

^{See 2.7.}

b)

^{If i is}

sufficiently large

^{such that}

qi

^>

zi

^{we rewrite}

Bi

in the form

where

II Bi3 ^II

⁼

z.. Furthermore, let ~ 1 _{) k. =}

and rewrite

B’. in

^{the form}

B. = Bi2

^with

^II Bi2

Thus we have obtained a

representation

where

Bil

^{is either empty or has a last}

^digit

different from

g-1,

^and

Bi2

^{is either}

^empty

^or consist of

(g-1 )’s only.

In this notation the transition from

p~ to p~ + j~ changes

^{the block}

B. into

^a block of the form

with the

following properties:

^II

B*i3

^{II = II}

^Bi3

^{II ,}

according

^{as to whether or not the}

inequality

holds

g

= 10, B; = 6089995301, j ~ i = ^4711, k, = ^3, z; = ^4, 6090000012, B. i1

⁼

^608, B.2 i

⁼

^999, B. i3

⁼

^5301,

=

609, B~ i2

⁼

^000, B.~ i3

^{= 0012.}

c)

^{If the n-th}

^digit

^{of a occurs} within the block

B;,

^{i.e. if n}

si,

^{then the}

«corresponding»

digit

^of

a*

has the index n’ for which

s;

^{- n’ =}

^{s; -}

^{n. In}

particular,

^{we have n’ > n for all n, and the}

image

^set

of

fN*

under this

mapping

^has

density

one, its

complement being

^{contained in the set}

s~

⁺

^1, s2

⁺

1,....

d)

^The

representation .

induces an obvious

decomposition

^{of the set}

flU*

into three

disjoint

^sets

11, I 2

^and

)~ ^where Ik

consists of those indices whose

position

^is

occupied by

^one of the blocks ^w

(k -1,2,3).

^{We need the}

following

^two

lemmas.

Proof. From the definitions of

qi and ji

^{we have :}

Hence,

^{if n is}

sufficiently large

^{and n}

si,

^{it follows that}

where the relation

(5)

has been used.

c p

Proof. If e >0 is

given,

^{we choose an}

integer

^{C >0 such that}

^{c .}

I2

^{be the set of those n}

~ , , , , , g p ^~

which are associated with one of the last C

digits

^{in some block Then}

Ix2

consists of chains of

lengths

^C

of consecutive

integers

each of which lies

strictly

^{between two} consecutive elements of S.

Hence,

^for any ⁿ

and

thus, by (5),

Let

I2l = 12 1 ( 2 Q,

then every ^{m E} is an index at which in the

expansion (1 )

^a copy of the block

(1) _G2 begins.

Therefore,

it follows from

(2)

^that

for all

sufficiently large

^n.

(~)

^{See 2.3.}

The assertion is

implied by (10) and (11 )

^since

e)

^{In order to establish}

normality of a* _by

means of the

equations analogous

^to

(2)

it suffices to let ⁿ tend to _*’

infinity through

the elements of _*

IN*’ _{(see step (c)).}

We consider a fixed block _* F ~=

F~

^{and an index}

n’ ^E

and

^we subdivide the

AF(a ^{,n’) copies}

^{of F among} the first n’

digits

^{of a} into three subsets :

1)

^Let

A1 (a*,n)

be the number of

copies occurring within,

^{but not at} the end of some block

B.. ^By

^virtue

of

(9),

^these

copies correspond

^to

copies

of F in the blocks

Bil.

^Hence

2)

^{The number}

Ar-(o! ,n’)

^of

^copies

of F which occur at the end of some block

8.1 ^(for

^these

^copies

^there

is ^no

corresponding

copy of F in the

expansion (1))

is bounded

by S* _(n’)

⁺

1,

^there

being

^{at most one such} copy in each block

B.. ^{Hence, by (5),}

3)

^{Each of the}

remaining copies

^{of F}

overlaps

^{with at least one} of the blocks

Bi2

^or

Bi3. ^Hence,

^{their num-}

ber is

and thus it follows from Lemmas 1 and 2 that

By combining (12), (13)

^and

(14)

^{we obtain}

and

therefore, by (2),

^{we have}

Adding

^these

inequalities

for all blocks F of the same

length

q and

observing

^that

we find that

equality

^{must hold in}

(15)

^for every F. A

simple

argument ^{shows now that the same}

equations

^are

valid for the lower

limits,

and thus the assertion follows.

4. REMARKS. ,

4.1. - The assertion we have

proved

^{ceases to hold if}

normality

^is

replaced by simple normality.

^For

example,

^let

g > 2, B. _ (g-2)~~) (g-3)~~)

^...

^0~~) (g-1 )~~) (i

⁼

1,2,...), j j~

^{= 1 for all .. i. Then the}

^integers

~B~) i

^are

uniquely determined, satisfying _{p 1 p 2}

.... Inasmuch as every block

B. contains ^{i zeros,}

ⁱ

ones etc.., the number a

= . B~ B~

^{... is}

^simply

^{normal to base g. But the}

number a * = g . ^{Bi B;}

^{... involves the}

blocks

A

simple

calculation shows that

Hence,

the number

a*

fails to be

simply

^normal.

4.2. - The

example just given

may be altered in such a way that the number ^a does not even possess

an

asymptotic distribution,

^{as we} shall demonstrate in the case g ⁼ 2.

Indeed,

^let

and define blocks

Bi (i =1,2,...)

^as

Writing again p; = 0 _(B.),

the numbers

p.

^are

uniquely

determined

(as

each block

B. ^begins

^{with the}

^{digit 1),}

and

they

^{form a} monotonic sequence since II

Bl ^II

^II

B2

^{II ....}

^Hence,

the number a

= 2 . Bl B2

^{... satisfies}

the conditions of Theorem

1,

except that instead of

being normal,

^{it is}

simply

^normal

only.

For the modified number

a~ . B* B;...

^{one has}

)

Now, let 03A3 k! = nj (j=1,2,...) and

k=1

These parameters

satisfy

^the

asymptotic equations

and

It will be convenient to rewrite the number

ain

the form

where

Each of the blocks

f 1 f 2

^...

f

^{, in}

⁽¹⁶⁾

ends with the sub-block 1

(2j) ; ^hence,

^the

corresponding

block in the ex-

pansion

^of

a*, consisting

^{of the}

first ^{m2j digits,}

^{ends with a sub-block}

^Q2j

in which all the

B~ ^satisfy

^{the second}

alternative of

(17),

^thus

having A~

⁼

2. ^2~

ⁱ

Therefore,

and thus

Consequently, using (18)

^and

(19),

^{we obtain}

On the other

hand,

^{a similar}

argument

^{shows that}

where the first alternative of

(17)

^{has been}

applied.

Therefore,

Since the

opposite inequality

^{is a} trivial consequence of

(17),

^{we have}

*

which shows that no limit

frequencies

^{for the}

digits

^{of a* exists.}

4.3. - Another

modification

of the

example given

in 4.1. demonstrates that the number a can be so

chosen that it remains

simply

^{normal to} base 2 but

a*

has

given

lower and upper

asymptotic digit frequencies, subjet only

^to the condition

which is

prompted by

the fact that a is

simply

normal and the

asymptotic frequency

^{of zeros cannot} be decreased

by

the transition from a to

a*.

We restate this

proposition

^{as the}

following

^theorem.

THEOREM 2. Given real numbers _r~ and

~’ satisfying

there exists a sequence

pi

^{-~ ~} of natural numbers such that the number a constructed to base 2 in the sense of

(l)

^is

simply

^normal

but, letting (1 ) j. i =1

^{for all}

^i,

the number

a*

satisfies the

equations

Proof. We consider the same

number 7

^as defined in

(16)

^{but we}

change

the definition of the blocks

Bi

^{as fol-}

lows :

for j =1,2,...

^{we let}

and if

nj’

^{we define}

Then the

integers _pi

⁼

~3 (B.)

^are

^easily

^{seen to}

^satisfy

the condition

Pi i -~ ~

and the number a

= . 2 B1

¹

B~ ...

is

simply

normal since

The modified blocks are

(1)

^{The theorem is also true for}

^arbitrary sequences ji

^with

log ji

^{= 0}

^(log pi)

^{but the}

^proof

^{would be more com-}

pi icated.

A 0 (a ~m)

Using again

the notation

(20)

^{we find}

by

^a

straight-forward

calculation that the lower limit lim is

~

^m

obtained as m runs

through

^the

ending positions

of the blocks

Q2~ (j =1,2,...).

These values of m

correspond

^to

the

end-points

^of

long

chains of blocks for which the first alternative of

(21 )

^is

valid,

^{and since}

in this _case, the result is

Similary, using

the second alternative of

(21 )

and the blocks thus

having

one obtains

4.4. - The result

just proved

^{can be}

generalised

^{to an}

arbitrary

^base g. Instead of

doing

^{this we shall}

replace

the condition that the number ^a be

simply

^normal

by requiring

^{that a and}

a*

have

given digit frequencies.

THEOREM 3. Let

g >

^{2 be an}

integer

^{and let}

g-adic digit frequency

^vectors

(1) 03BE

⁼

(03BEo,03BE1,

^,

^{..., 03BE} _ )

^,

§

^_

(03B6*o, 03B6*1 , ^{..., 03B6} g-1 )

^be

^given.

Then there exists a sequence

p2

^{... such}

that, letting ( ) j _i

_; ^{=1 for}

all i the numbers a and

a*

in the sense of Theorem 1 have

digit frequencies ~’

^and

~’ *

^{if and}

^only

^if

(3)

COROLLARY. ^. The

equations (23) imply

the relation

Proof. a.l. Assume that numbers a and

a*

whith the stated

properties

exist. In order to compare with

A (B.) (i = 1,2,...)

^we

distinguish (in

the notation of section

3)

^the

following

^{five cases}

(4) :

1 ^*

g-l g-l

(1 )

I.e. real numbers

~’k =1 .

k=0 k=0

(2)

^{The theorem is also true for}

^arbitrary sequences ji

^with

^{log ji}

^{= 0}

^(log p;)

^{but the}

^proof

^{would be more com-}

plicated.

(3)

^{It should be observed}

^that,

^{for g =}

^{2, (24)}

^{is trivial and}

⁽²³⁾

^{is void.}

(4) ^Observing

^{that in case 3 one has}

a(B.)

⁺

p(Bil)

^II

Bi

^{II , we find that} this list of cases is

complete

^and

non-overlapping.

^For g ⁼

2,

^{cases 1 and 4 do not occur.}

On

modifying

constructed

1) Q (Bi) = 0 , p (Bi) = 0 (example : 19805)

2) a (Bi) = 0 ,

^{p >0}

(example : 19800)

3) Q(Bi) ^{>0 ,}

^P

(Bil )

^{> 0}

(example : 900999 ;

⁼

900) 4) II Bi

^{II >}

a (Bi)

^>

^0,

⁼⁰

(example : 108999, Bi~

⁼

¹⁰⁸⁾

5)

^II

B~

^{II =}

a(Bi) (example : 99999).

Computing

the blocks

B = ^{~i(~i ~} (Bi)

⁺

^{1 )}

^{we obtain}

Furthermore,

^{we find}

A

simple

argument,

using

the relation II

B* i

^{II =}

q.~ i -~ ~ (see (4))

^to handle the terms -1 in case

2,

^{now shows}

that

Hence,

^condition

(22)

is necessary.

~y.2. - ^- To _prove the

necessity

^of

(23)

^we may ^assume that _g > 2. Let k be a fixed

digit

^with

1 k

g-2. Then, considering again

^{the five cases} introduced in

a.1),

^{we find}

always

^that

Ak (Bi)

^{= 0}

or ±

1,

from which the

equations (23)

^can be deduced.

b.

Conversely,

^{let us assume that}

vectors 03B6 and 03B6*

^{with the stated}

properties

^be

given.

^{Then we}

construct a

pair

of numbers _a,

a by modifying

^the construction of Theorem 2 as follows : we now

introduce,

^for

j

⁼

1,2,...,

^{the functions}

noting that,

in view of

(24),

If nj-1 i nj , we define the block Bi as

letting again

^a

= g . B~ B~

^...,

B~= (B.)

⁺

^1),

^and

It is _easy to show that the number a has the

digit frequency vector § , using

^{the relations II}

Bi

^II

^{= j -~}

^oo,

i (h = 0,...,g-2)

^and

Uj

⁺

( ~’ g-1 * + ~ ~ - ~ 0 ^o)j = ~ - g 1 j,

^{the last one}

^being

^obtained

^by

^means

of

(24).

Similary,

^{inasmuch as}

an obvious

computation, using (27)

and the relation

* *

proves that the number a has the

given digit frequency vector t

^.

5. GENERALISATION

Theorem 1 _may be

interpreted

^as

stating

^{that if the number a has}

Lebesgue

^{measure X as its}

g-adic

*

distribution,

^{the same is true for a .}

Hence,

^one may ask whether there are distribution measures other than B

having

^{the same} invariance

property.

^This

question

^{is answered}

by

^the

following proposition.

THEOREM 4. Let x be a

g-adic

distribution measure. Then _any number c~ constructed ^as in Theorem 1 and

having

^the

g-adic

distribution

(1 )

^{x , , will}

yield

^{a number}

a*

with the same distribution if and

only

^if

(2)

x

( 1 ~ ) = ^0,

i.e. if the

point

^{1 is not an atom of x .}

(1 ) ^{) I.e.,}

^{for every}

^{block F,}

^{lim =}

^{x (F)}

^{in the sense of 2.6.}

o n ⁿ *

( )

^It should be noted that this condition is violated

by

^the

examples

discussed in section 4 whenever c~ and a

have different distributions.

Proof.

a)

^{Let x be a}

given g-adic

distribution _measure, i.e. ^a. Borel

probability

measure on

[0,1 ] which

^is

invariant under the

g-adic

^shift operator

T g

^x

= g x ~ , satisfying

^the

assumption x ( ~ 1 ~ )

^{= 0.}

Parts

a), b), c)

^and

d)

^{of the}

proof

of theorem 1 remain

valid,

except that in the

proof

^{of Lemma 2}

the

following change

^{is now} necessary. Since

(~ ) the

intersection of the basic intervals

G Q

^is

the

assumption x ( { 1 } )

^{= 0 is}

equivalent

^to

Hence,

^{if e > 0 is}

given,

there exists an Q such that x

(GQ)

^{e .} The remainder of the

proof

^is

unchanged,

^ex- cept ^{that the}

inequality (11 )

^{has to be}

replaced by

The fact that

a*

has distribution x is then established

by

^the argument ^of part

e)

^{in the}

proof

of Theorem

1,

1

with the term ^- in

(15) being replaced by x (F).

b) Conversely,

^{let x be a}

g-adic

distribution measure with x

({ 1})

^{= ~} > 0. We shall show that there exists a sequence

pi

^with

pi

^{l’ oo such}

that,

^{in the sense of the}

theorem,

^a has the distribution x but

a*

does not.

b.l. ^- As the first step ^{we consider a number}

a1

with the distribution x as constructed

(2) ) by J.

^VILLE

[6], following

^the

exposition

of A.G. POSTNIKOV

[5], Chapter

^{III. Given} any sequence

e

1 >

e 2 ^>..., E k ~ ^0,

this construction defines

recursively

^blocks

Lo,

^{..., in such a way}

^{that, letting}

and II

Lo L~

^...

Lk

^{II =}

nk,

^any block F =~

F~,

^{II F II}

^k,

satisfies the

inequalities

for all n >

nk,

^The

length

^II

Lk

^{II may, at each stage of the}

construction,

^{be chosen}

arbitrarily large

in relation to the II

!!. Hence,

^we may choose each

nk+1 (k= 1,2,...)

^so

large that,

for all F with II F II

k,

(~)

^{See 2.3 and 2.6.}

(2) ⁾

^An

^elegant

construction of such

numbers, using graph theory,

^was

recently given by

^H.

KUHNLE _{[4] .}

For the same reason we may

furthermore

assume that

Furthermore,

^we

require

that the first

digit

^of

Lo

be different from 0.

6.2. -

By (29)

^{we have}

Hence,

^an

application

of Dirichlet’s drawer

principle

shows that for each k there exists an arithmetic

progression

{ ^{rk’ rk +k,} rk +2k, ...}

^{, 1}

^k,

which contains more than

1 k R(k) numbers

^{n such that some} copy

of G

ends at the n-th

digit

of the block

Lk+ 1.

Let be the block obtained from

(k

⁼

0,1,2,...) by

^a

cyclic

^shift

by k-rk ^places

^{to the}

right.

I n view of

(30)

^{it is}

possible

^to rewrite each block

(k = 1,2,...)

in the form

where

II

₁ ^{II = ... =}

II Y k h

^{II = k and at least -}

^R(k)

of these blocks are

equal

^to

Gk,

^to be called relevant

k k

copies.

Now,

^{let a number}

a2

be defined as

It is _easy to see that

a2

also has the

digit

distribution x .

Indeed,

^{if ~denotes the set of}

places occupied

^{in the}

g-adic expansion

^of

a~ ^by

^{the first and the last}

^digit

^of

L~,

^{the first two} and the last two

digits

^of

L2,etc,

^then

it is evident

that,

^{if a} block F with II F II ⁼ k > 0 is

given,

the transition from

c~i

^to

a2

^can

only

^{create or}

destroy copies

^{of F to the}

right

of the index

nk

^if

they begin

^{or terminate at}

places belonging

^to

Hence,

^we

have,

^for all n

nk,

and

since, by (31 ),

⁼

o(n),

^it follows that

i.e.

a2

^{has the} distribution x .

b.3. - The blocks

(k = 1,2,...)

^are

changed

^{as follows : we} define blocks

thus

replacing (33) by

and then

letting a3 = g . L’2 L3 L4 ..,

^.

The first

digits

of all blocks

Yki

^{determine a set V}

= ~ ^Vi, v2, ... ~

of indices for which

vm = k

^whenever

the index

vm

^occurs within the block

Lk. ^{Hence, by (31 ), V(m)}

⁼

^o(m)

^{and thus}

a3 ^again

^{has the}

^digit

^distribu-

tion x .

Furthermore,

it follows from the definition of 1 that

(with (~k+1 ) - ^{1 2).}

6.4. - ^-

Finally,

^we renumber the blocks

Yki ^{(k = 1,2,... ;} i=1,...,hk)

^without

changing order,

^as

B1,B2,w,

^thus

rewriting a3

^{as a}

= g . B~ B2

^.... Then the number a satisfies the

assumptions

of the theorem since :

1)

it has the

distribution x , 2)

^{none of the}

blocks Bi ^begins

^with

^0,

^{hence the}

integers pi = 03B2-1 _(Bi)

^are

uniquely determined, 3) _Pi

^{as II}

Bi

^II

Furthermore,

each relevant _copy of a block

Gk (k = 1,2,...)

forms one of the blocks

Bi.

We shall also use the fact that the

assumption log ji

⁼

^o(log p~) ^implies

the relation

~ ~ ~

~.3. - For the

corresponding

^{number o;}

= . Bi B~ ...

^{we also use} the block de

composition

where !s obtained from

L. by subjecting

each of its blocks

B.

^{to the}

* operation

of Theorem 1. Each relevant copy

of a block ^G.,

which forms a block

B,

with k >

z;,

^{furnishes a block}

(see (6))

of the form

B*i ⁼

^{1 0}

’

Bi3 ,

^!!

B.o

^{!! =}

z.,

^where

Bi3

is defined as in the

proof

of Theorem 1. We shall show that 03B1

* has

«much fewer»

(g"1)’s

^{than a.}

Indeed, letting

^we

have, if B;

is embedded in with

Hence, applying (37)

^{to each of} the relevant

copies

^of

Gk

^and

^observing

^that

^they

^{are contained in}

^(and

therefore,

^{in we}

have,

^{for k}

=1,2,... ,

Since it follows from the definition of that

this implies

Letting

^{k tend to}

infinity, observing

^that

E k -~ ^0,

^{0 r~ = lim x}

(Gk)

^x

(G1 ), k1

^{= k}

^{by (36),}

- k-~~

k ⁼

o(

^II

^II ) by (31 )

^{and II II > II I)}

by definition,

^we obtain the relation

"

II

x ^_

Therefore,

^lim 201420142014201420142014 *

~(g-1) -

^r~

x (g-1 ),

and thus a

*

does not have the distribution x .

noo ⁿ

Remark. Theorem 4 may be

applied

^{to show}

that,

^{in the sense of Theorem}

2,

^for any k > 1 there exists even a k-normal number a such that

a

fails to be

simply

normal. It suffices to consider _any

g-adic

distribution measu- re x such that x

(F) =

^{for all F with}

^II

^F

^II

⁼

k,

but with x

({1 j)

^>

^{0, and}

to construct a

pair

of

numbers a,

^a

by

the method of Theorem 4. For

example,

^if g ⁼

2,

^{k =}

2,

^{we can choose}

where

bx

is the Dirac measure concentrated at x. It is

easily

^{seen that the measure x} is invariant under the

binary

^shift operator

T2,

^{and that any}

corresponding

^{number a} is 2-normal but

a*

is not.

285

REFERENCES

[1]

^D.G. CHAMPERNOWNE. «The construction of decimals normal in the scale of ten».

Journ.

^London

Math. Soc. 8

(1933),254-260.

[2]

A.H. COPELAND and P.

ERDÖS.

«Note on normal numbers». Bull. Amer. Math. Soc. 52

(1946),

857-860.

[3]

^{H. DAVENPORT and P.}

ERDÖS.

«Note on normal decimals». Canad.

Journ.

^{Math. 4}

(1952),

^58-63.

[4]

^H.

KÜHNLE. «Schnell konvergierende Ziffernfolgen bei speziellen zahlentheoretischen

Transforma- tionen». Diss. Univ.

Stuttgart

^1978.

[5]

A.G. POSTNIKOV.

«Ergodic problems

^{in the}

theory of congruences and of diophantine approxima-

tions».

(Russian). Trudy

Mat. Inst. Stekov 82

(1966). Engl.

transl. Proc. Stekov Inst. Math.

82,

^Amer.

Math. Soc. 1967.

[6] J.

VILLE. «Etude

critique

^{de la notion} de collectif».

Monogr.

^des

Probabilités, 3, Gauthiers-Villars,

Paris 1939.

(Manuscrit

reçu le 19 mai

1979)